I was actually thinking of this as well. I thought of a prusik with hook on each tie-out. That way, you go around the tree, with the prusik on the free end (i.e. between the tree and the loose end of the tie-out). Use the hook to attach to the tarp. If you do that on both sides, it will then be possible to centre the tarp properly by moving the prusiks.
It's really only one long line with the prusik and hook on each end. You could set it up first and then drape the tarp over it (I guess you could add drip lines too in this case). So long as the tarp has something to clip onto, you're all set!